Robust single-item lot-sizing problems with discrete-scenario lead time

In this paper, we consider the minmax robust lot-sizing problem (RLS) with uncertain lead times. We explore the uncapacitated (RULS) and capacitated (RCLS) cases with and without lost sales, with discrete scenarios. We provide a complexity analysis proving that robust lot sizing problems are NP-hard even in the case with two scenarios. For each case, a mixed integer programming model is proposed. We show that several optimality conditions for the deterministic cases provided in Wagner and Whitin (1958), Aksen et al. (2003), as well as a classical facility location based model, are no longer valid. However, we generalize these properties to the lot sizing problem with deterministic but dynamic lead times and demonstrate their validity. Numerical experiments on instances with up to 400 periods and 25 scenarios reveal that, against all expectation, the capacitated version of the robust lot sizing problem is easier to solve than the uncapacitated version.